Contributions to global optimization using interval methods and speculation

Abstract

Most electronic devices we are familiar with, such as cell phones and computers, are small and require similarly small electronic components arranged and connected in small areas. Finding the right size and arrangement of the components inside a device can be a challenge. The manufacturing process of the components limits their possible size, some components have specific needs to operate at a certain speed, and the total area of the device is also limited. In portable devices, these designs have one important objective: that the entire device consumes the minimum amount of electricity possible, so the device can keep functioning for a longer time without recharging its battery. Engineers could try multiple designs and see which one is best. This unstructured approach would be inefficient, require a lot of time, and still likely not guarantee that the best configuration has been identified. ^ Instead, we can express the design restrictions as a series of mathematical equations, the parameters of the design as a set of variables, and the energy as an objective function to be minimized. This is an optimization problem, a category of problems represented as mathematical models in which we seek the minimum (or maximum) value of an objective function, while possibly meeting some constraints / requirements. ^ In order to solve an optimization problem, search algorithms are needed. Local search focuses on making slight adjustments to the values of the parameters to get progressively better objective function values, and stopping the search when the objective value cannot be improved by any nearby parameter values. Local search algorithms rely on an initial "guess" of the parameter values to converge to the closest minimum of the objective function. Because of this, they cannot guarantee that the solution is the overall minimum (a global optimum). Global search uses techniques that expand the range of possible values the parameters to overcome the drawback of local search. Global search techniques trade off execution time for improved accuracy, taking a longer time to find the global optimum. ^ In this thesis, a global optimization search algorithm is introduced. The main goal of this algorithm is to guarantee an objective function value that is a global optimum. Interval are used to model the search space and guarantee an exhaustive search, hence a global result. ^ Two search algorithms were developed. The first algorithm uses speculation over the interval range of the objective function, by placing "bets'" on the expected minimum value until finding it or proving it is not correct. The second algorithm is an enhancement of the first one, improving the convergence of the upper bound of the objective function by using derivative-based local search techniques for constrained and unconstrained optimization. ^ Our results show promise. The first algorithm finds good interval enclosures of the objective function for both constrained and unconstrained problems. The second algorithm shows improvements on unconstrained optimization by finding a better upper bound of the objective function's evaluation interval in less time than the first algorithm. These contributions will help in developing more efficient and guaranteed global optimizers for complex optimization problems. ^